In order for Skype to work, your phone must be on the internet somehow. Obviously WiFi will do that but also if you buy a local SIM card in Europe, you could get one that includes some data, too. Mobile data with a local SIM puts you on the internet without WiFi. (But with this option, your phone must be unlocked, as I said earlier.) If you replace your AT&T SIM card with a local SIM card, then no AT&T roaming would apply.
If you don't go the local SIM route, then you would still have your AT&T SIM installed. It sounds like, if you don't pay for an international package with AT&T, then your phone may simply not work at all in Europe - which may be what you want to avoid the costs - but if it still registers with a local mobile network, you could go into settings on the phone and simply take it off, I believe.
And if you don't go the local SIM route - yes, you'd be limited to WiFI-only use of the phone with Skype for voice, with other services for texting.
Airplane mode disables mobile networks, data, and perhaps WiFi, but you can then enable WiFi separately while still in airplane mode, I believe. Then you'd be able to use Skype.
You can install the Skype app for free on your phone right now. The free option doesn't let you call phones, only other phones or computers that are on Skype. You can try this: install Skype (includes setting up an account), then put the phone into airplane mode, then enable WiFi. Then try to call your daughter (not via a phone number - "call" her on her computer or her Skype-enabled phone). Connecting to someone on Skype not using a phone is free but also a tad more complicated; you must have them add you as a contact first, but your daughter can probably help you with that.
If you get the above to work, then try using Skype again using your phone's data plan and not WiFi. This will simulate how it will work in Europe if you buy a local SIM card and use its data plan for Skype calls.